Abstract: Advanced models for quantum computation where even the circuit connections
are subject to the quantum superposition principle have been recently
introduced. There, a control quantum system can coherently control the order in
which a target quantum system undergoes $N$ gate operations. This process is
known as the quantum $N$-switch, and has been identified as a resource for
several information-processing tasks. In particular, the quantum $N$-switch
provides a computational advantage -- over all circuits with fixed gate orders
-- for phase-estimation problems involving $N$ unknown unitary gates. However,
the corresponding algorithm requires the target-system dimension to grow
(super-)exponentially with $N$, making it experimentally demanding. In fact,
all implementations of the quantum $N$-switch reported so far have been
restricted to $N=2$. Here, we introduce a promise problem for which the quantum
$N$-switch gives an equivalent computational speed-up but where the
target-system dimension can be as small as 2 regardless of $N$. We use
state-of-the-art multi-core optical fiber technology to experimentally
demonstrate the quantum $N$-switch with $N = 4$ gates acting on a
photonic-polarization qubit. This is the first observation of a quantum
superposition of more than 2 temporal orders, and also demonstrates its
usefulness for efficient phase-estimation.